page_title: Structure tensor
section_title: Interpretation
hierarchical_section_title: Structure tensor / The 3D structure tensor / Interpretation
caption_attribution_description: English: Ellipsoidal representation of the 3D structure tensor
context_page_description: In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent. The structure tensor is often used in image processing and computer vision.
context_section_description: As in the three-dimensional case, the eigenvalues  of , and the corresponding eigenvectors , summarize the distribution of gradient directions within the neighborhood of p defined by the window .  This information can be visualized as an ellipsoid whose semi-axes are equal to the eigenvalues and directed along their corresponding eigenvectors.
  In particular, if the ellipsoid is stretched along one axis only, like a cigar (that is, if  is much larger than both  and ), it means that the gradient in the window is predominantly aligned with the direction , so that the isosurfaces of  tend to be flat and perpendicular to that vector.  This situation occurs, for instance, when p lies on a thin plate-like feature, or on the smooth boundary between two regions with contrasting values.
  If the ellipsoid is flattened in one direction only, like a pancake (that is, if  is much smaller than both  and ), it means that the gradient directions are spread out but perpendicular to ; so that the isosurfaces tend to be like tubes parallel to that vector.  This situation occurs, for instance, when p lies on a thin line-like feature, or on a sharp corner of the boundary between two regions with contrasting values.
  Finally, if the ellipsoid is roughly spherical (that is, if ), it means that the gradient directions in the window are more or less evenly distributed, with no marked preference; so that the function  is mostly isotropic in that neighborhood.  This happens, for instance, when the function has spherical symmetry in the neighborhood of p. In particular, if the ellipsoid degenerates to a point (that is, if the three eigenvalues are zero), it means that  is constant (has zero gradient) within the window.
caption: Ellipsoidal representation of the 3D structure tensor.
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